Points $A$, $B$, $C$, $D$, and $E$ are located in 3-dimensional space with $AB= BC= CD= DE= EA= 2$ and $\angle ABC = \angle CDE = \angle
DEA = 90^\circ$.  The plane of triangle $ABC$ is parallel to $\overline{DE}$. What is the area of triangle $BDE$?
Answer: In coordinate space, let $D = (0,0,1)$ and $E = (0,0,-1).$  Since $CD = EA = 2,$ $C$ lies on a circle centered at $D$ with radius 2, and $A$ lies on a circle centered at $E$ with radius 2.  Furthermore, $\angle CDE = \angle DEA = 90^\circ,$ so these circles lies in planes that are perpendicular to $\overline{DE}.$

[asy]
import three;

size(200);
currentprojection = perspective(4,3,2);

triple A, B, Bp, C, D, E;
real t;

A = (sqrt(3),1,-1);
B = (sqrt(3),-1,-1);
Bp = (sqrt(3),1,1);
C = (sqrt(3),-1,1);
D = (0,0,1);
E = (0,0,-1);

path3 circ = (2,0,-1);
for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) {
  circ = circ--((0,0,-1) + (2*cos(t),2*sin(t),0));
}

draw(circ);

path3 circ = (2,0,1);
for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) {
  circ = circ--((0,0,1) + (2*cos(t),2*sin(t),0));
}

draw(circ);
draw(C--D--E--A);

dot("$A$", A, S);
dot("$C$", C, W);
dot("$D$", D, NE);
dot("$E$", E, dir(0));
[/asy]

We can rotate the diagram so that $D$ and $E$ have the same $x$-coordinates.  Let $A = (x,y_1,-1)$ and $C = (x,y_2,1).$  Since $EA = CD = 2,$
\[x^2 + y_1^2 = x^2 + y_2^2 = 4.\]Then $y_1^2 = y_2^2,$ so $y_1 = \pm y_2.$

Furthermore, since $AB = BC = 2$ and $\angle ABC = 90^\circ,$ $AC = 2 \sqrt{2}.$  Hence,
\[(y_1 - y_2)^2 + 4 = 8,\]so $(y_1 - y_2)^2 = 4.$  We cannot have $y_1 = y_2,$ so $y_1 = -y_2.$  Then $4y_1^2 = 4,$ so $y_1^2 = 1.$  Without loss of generality, we can assume that $y_1 = 1,$ so $y_2 = -1.$  Also, $x^2 = 3.$  Without loss of generality, we can assume that $x = \sqrt{3},$ so $A = (\sqrt{3},1,-1)$ and $C = (\sqrt{3},-1,1).$

Finally, we are told that the plane of triangle $ABC$ is parallel to $\overline{DE}.$  Since both $A$ and $C$ have $x$-coordinates of $\sqrt{3},$ the equation of this plane is $x = \sqrt{3}.$  The only points $B$ in this plane that satisfy $AB = BC = 2$ are the vertices $B_1$ and $B_2$ of the rectangle shown below, where $B_1 = (\sqrt{3},-1,-1)$ and $B_2 = (\sqrt{3},1,1).$

[asy]
import three;

size(200);
currentprojection = perspective(4,3,2);

triple A, B, Bp, C, D, E;
real t;

A = (sqrt(3),1,-1);
B = (sqrt(3),-1,-1);
Bp = (sqrt(3),1,1);
C = (sqrt(3),-1,1);
D = (0,0,1);
E = (0,0,-1);

path3 circ = (2,0,-1);
for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) {
  circ = circ--((0,0,-1) + (2*cos(t),2*sin(t),0));
}

draw(circ);
draw(surface(A--B--C--Bp--cycle),paleyellow,nolight);

path3 circ = (2,0,1);
for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) {
  circ = circ--((0,0,1) + (2*cos(t),2*sin(t),0));
}

draw(circ);
draw(C--D--E--A);
draw(A--B--C--Bp--cycle);

dot("$A$", A, S);
dot("$B_1$", B, W);
dot("$B_2$", Bp, N);
dot("$C$", C, W);
dot("$D$", D, NE);
dot("$E$", E, dir(0));
[/asy]

In either case, triangle $BDE$ is a right triangle where the legs are both 2, so its area is $\frac{1}{2} \cdot 2 \cdot 2 = \boxed{2}.$